CHAP, ill] PARTITIONS. 155 



with or without repetitions. In Ch. 6, he gave an exposition of Euler's 

 work on partitions and Sylvester's theory of waves, illustrated by examples. 

 In Ch. 7 it is noted that any relation between two partitions of n leads to an 

 identity between two infinite series. 



A. S. Werebrusow 180a noted that if a, b, -,k,l are positive integers and 

 if {n} denotes the number of sets of positive integral solutions of 



/ = ax + by + + kt = n, 



the number of sets for/+ lu = m is {m 1} + {m 21} + {m 31} -\ ---- . 

 D. Gigli 181 considered the number N a of combinations of 1, , m taken 

 n at a time with the sum s. The least s is L = n(n + l)/2 and the greatest 

 is G = mn n(n l)/2. It is shown by induction that N L , N L+ i, , N G 

 are the coefficients of the powers of x in the expansion of 



(1 x m }(\ x m ~ l ) (!- x m ~ n+l ) 



C. F. Gauss 182 had treated this function without reference to partitions and 

 noted that 



m 1 



(m, n) = (m, m n), (m, ju + 1) = X) 3*~' t ft /*) 



i=H 



Gigli tabulated the N's for m = 10, n = 2, 3, , and proved that 



(m, n) = X) x n(p ~ l} (m p, n 1). 

 P =\ 



T. Muir 183 noted that there are C n -kr+k, r combinations of n elements 

 taken r at a time such that no element is taken along with any one of the k 

 elements immediately following it in the initial set. The number of sets 

 of r things obtained from n by omitting n r of them so chosen that they 

 form (n r}jk sets of lc consecutive things is C,, r , where s = (n + kr r)jk. 



E. Landau 184 discussed the maximum order of literal substitutions on 

 a given number n of letters. It is a question of the maximum of the l.c.m. 

 of Oi, , a v in all decompositions n = 0,1 + + a v of n into positive 

 integral summands. Cf. Landau. 196 



E. Netto 185 found the number of cyclic decompositions obtained by 

 arranging in a circle each of the (Jll) decompositions of n into p summands 

 wi.th attention to order. 



L. Brusotti 186 proved the result of Catalan's. 25 



F. H. Jackson 187 wrote P x for p\ l - - - p x and \j) x z] n for 



(1 -f P*+(-*g)(l + pgHn-^g) . . . (1 



P x ~ l z](l 



1800 Spaczinski's Bote, Odessa, 1901, Nos. 298-9, pp. 224-9, 250-4. 

 181 Rendiconti Circ. Mat. Palermo, 16, 1902, 280-5. 

 182 Comm. Soc. Getting., 1, 1811; Werke, II, 16-17. 



183 Proc. Roy. Soc. Edinburgh, 24, 1901-3, 102-4. 



184 Archiv Math. Phys., (3), 5, 1903, 92-103. 



185 Ibid., 185-196. 



186 Periodico di Mat., 17, 1903, 191-2. 



187 Proc. London Math. Soc., (2), 1, 1903-4, 63-88. 



